Theorem xlt2add 9867

Description: Extended real version of lt2add 8392. Note that ltleadd 8393, which has weaker assumptions, is not true for the extended reals (since 0 + +oo < 1 + +oo fails). (Contributed by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
xlt2add	|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> ( ( A < C /\ B < D ) -> ( A +e B ) < ( C +e D ) ) )

Proof of Theorem xlt2add

Step	Hyp	Ref	Expression
1		xaddcl 9847	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) e. RR* )
2	1	3ad2ant1 1018	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e B ) e. RR* )
3	2	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e B ) e. RR* )
4		simp1l 1021	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> A e. RR* )
5		simp2r 1024	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> D e. RR* )
6		xaddcl 9847	. . . . . . . 8 |- ( ( A e. RR* /\ D e. RR* ) -> ( A +e D ) e. RR* )
7	4, 5, 6	syl2anc 411	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e D ) e. RR* )
8	7	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e D ) e. RR* )
9		xaddcl 9847	. . . . . . . 8 |- ( ( C e. RR* /\ D e. RR* ) -> ( C +e D ) e. RR* )
10	9	3ad2ant2 1019	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( C +e D ) e. RR* )
11	10	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( C +e D ) e. RR* )
12		simp3r 1026	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> B < D )
13	12	adantr 276	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> B < D )
14		simp1r 1022	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> B e. RR* )
15	14	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> B e. RR* )
16	5	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> D e. RR* )
17		simprl 529	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> A e. RR )
18		xltadd2 9864	. . . . . . . 8 |- ( ( B e. RR* /\ D e. RR* /\ A e. RR ) -> ( B < D <-> ( A +e B ) < ( A +e D ) ) )
19	15, 16, 17, 18	syl3anc 1238	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( B < D <-> ( A +e B ) < ( A +e D ) ) )
20	13, 19	mpbid 147	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e B ) < ( A +e D ) )
21		simp3l 1025	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> A < C )
22	21	adantr 276	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> A < C )
23	4	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> A e. RR* )
24		simp2l 1023	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> C e. RR* )
25	24	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> C e. RR* )
26		simprr 531	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> D e. RR )
27		xltadd1 9863	. . . . . . . 8 |- ( ( A e. RR* /\ C e. RR* /\ D e. RR ) -> ( A < C <-> ( A +e D ) < ( C +e D ) ) )
28	23, 25, 26, 27	syl3anc 1238	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A < C <-> ( A +e D ) < ( C +e D ) ) )
29	22, 28	mpbid 147	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e D ) < ( C +e D ) )
30	3, 8, 11, 20, 29	xrlttrd 9796	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e B ) < ( C +e D ) )
31	30	anassrs 400	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) /\ D e. RR ) -> ( A +e B ) < ( C +e D ) )
32		pnfxr 8000	. . . . . . . . . . . 12 |- +oo e. RR*
33	32	a1i 9	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> +oo e. RR* )
34		pnfge 9776	. . . . . . . . . . . 12 |- ( C e. RR* -> C <_ +oo )
35	24, 34	syl 14	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> C <_ +oo )
36	4, 24, 33, 21, 35	xrltletrd 9798	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> A < +oo )
37		npnflt 9802	. . . . . . . . . . 11 |- ( A e. RR* -> ( A < +oo <-> A =/= +oo ) )
38	4, 37	syl 14	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A < +oo <-> A =/= +oo ) )
39	36, 38	mpbid 147	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> A =/= +oo )
40		pnfge 9776	. . . . . . . . . . . 12 |- ( D e. RR* -> D <_ +oo )
41	5, 40	syl 14	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> D <_ +oo )
42	14, 5, 33, 12, 41	xrltletrd 9798	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> B < +oo )
43		npnflt 9802	. . . . . . . . . . 11 |- ( B e. RR* -> ( B < +oo <-> B =/= +oo ) )
44	14, 43	syl 14	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( B < +oo <-> B =/= +oo ) )
45	42, 44	mpbid 147	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> B =/= +oo )
46		xaddnepnf 9845	. . . . . . . . 9 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo )
47	4, 39, 14, 45, 46	syl22anc 1239	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e B ) =/= +oo )
48		npnflt 9802	. . . . . . . . 9 |- ( ( A +e B ) e. RR* -> ( ( A +e B ) < +oo <-> ( A +e B ) =/= +oo ) )
49	2, 48	syl 14	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( ( A +e B ) < +oo <-> ( A +e B ) =/= +oo ) )
50	47, 49	mpbird 167	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e B ) < +oo )
51	50	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = +oo ) -> ( A +e B ) < +oo )
52		oveq2 5877	. . . . . . 7 |- ( D = +oo -> ( C +e D ) = ( C +e +oo ) )
53		mnfxr 8004	. . . . . . . . . . 11 |- -oo e. RR*
54	53	a1i 9	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo e. RR* )
55		mnfle 9779	. . . . . . . . . . 11 |- ( A e. RR* -> -oo <_ A )
56	4, 55	syl 14	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo <_ A )
57	54, 4, 24, 56, 21	xrlelttrd 9797	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo < C )
58		nmnfgt 9805	. . . . . . . . . 10 |- ( C e. RR* -> ( -oo < C <-> C =/= -oo ) )
59	24, 58	syl 14	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( -oo < C <-> C =/= -oo ) )
60	57, 59	mpbid 147	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> C =/= -oo )
61		xaddpnf1 9833	. . . . . . . 8 |- ( ( C e. RR* /\ C =/= -oo ) -> ( C +e +oo ) = +oo )
62	24, 60, 61	syl2anc 411	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( C +e +oo ) = +oo )
63	52, 62	sylan9eqr 2232	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = +oo ) -> ( C +e D ) = +oo )
64	51, 63	breqtrrd 4028	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = +oo ) -> ( A +e B ) < ( C +e D ) )
65	64	adantlr 477	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) /\ D = +oo ) -> ( A +e B ) < ( C +e D ) )
66		simpr 110	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = -oo ) -> D = -oo )
67		mnfle 9779	. . . . . . . . . 10 |- ( B e. RR* -> -oo <_ B )
68	14, 67	syl 14	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo <_ B )
69	54, 14, 5, 68, 12	xrlelttrd 9797	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo < D )
70		nmnfgt 9805	. . . . . . . . 9 |- ( D e. RR* -> ( -oo < D <-> D =/= -oo ) )
71	5, 70	syl 14	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( -oo < D <-> D =/= -oo ) )
72	69, 71	mpbid 147	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> D =/= -oo )
73	72	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = -oo ) -> D =/= -oo )
74	66, 73	pm2.21ddne 2430	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = -oo ) -> ( A +e B ) < ( C +e D ) )
75	74	adantlr 477	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) /\ D = -oo ) -> ( A +e B ) < ( C +e D ) )
76		elxr 9763	. . . . . 6 |- ( D e. RR* <-> ( D e. RR \/ D = +oo \/ D = -oo ) )
77	5, 76	sylib 122	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( D e. RR \/ D = +oo \/ D = -oo ) )
78	77	adantr 276	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) -> ( D e. RR \/ D = +oo \/ D = -oo ) )
79	31, 65, 75, 78	mpjao3dan 1307	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) -> ( A +e B ) < ( C +e D ) )
80		simpr 110	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = +oo ) -> A = +oo )
81	39	adantr 276	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = +oo ) -> A =/= +oo )
82	80, 81	pm2.21ddne 2430	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = +oo ) -> ( A +e B ) < ( C +e D ) )
83		oveq1 5876	. . . . 5 |- ( A = -oo -> ( A +e B ) = ( -oo +e B ) )
84		xaddmnf2 9836	. . . . . 6 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
85	14, 45, 84	syl2anc 411	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( -oo +e B ) = -oo )
86	83, 85	sylan9eqr 2232	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = -oo ) -> ( A +e B ) = -oo )
87		xaddnemnf 9844	. . . . . . 7 |- ( ( ( C e. RR* /\ C =/= -oo ) /\ ( D e. RR* /\ D =/= -oo ) ) -> ( C +e D ) =/= -oo )
88	24, 60, 5, 72, 87	syl22anc 1239	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( C +e D ) =/= -oo )
89		nmnfgt 9805	. . . . . . 7 |- ( ( C +e D ) e. RR* -> ( -oo < ( C +e D ) <-> ( C +e D ) =/= -oo ) )
90	10, 89	syl 14	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( -oo < ( C +e D ) <-> ( C +e D ) =/= -oo ) )
91	88, 90	mpbird 167	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo < ( C +e D ) )
92	91	adantr 276	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = -oo ) -> -oo < ( C +e D ) )
93	86, 92	eqbrtrd 4022	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = -oo ) -> ( A +e B ) < ( C +e D ) )
94		elxr 9763	. . . 4 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
95	4, 94	sylib 122	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
96	79, 82, 93, 95	mpjao3dan 1307	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e B ) < ( C +e D ) )
97	96	3expia 1205	1 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> ( ( A < C /\ B < D ) -> ( A +e B ) < ( C +e D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 977   /\ w3a 978   = wceq 1353   e. wcel 2148   =/= wne 2347   class class class wbr 4000  (class class class)co 5869   RRcr 7801   +oocpnf 7979   -oocmnf 7980   RR*cxr 7981   < clt 7982   <_ cle 7983   +ecxad 9757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-i2m1 7907  ax-0id 7910  ax-rnegex 7911  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-xadd 9760

This theorem is referenced by:  bldisj  13568